What is Sine: Definition and 519 Discussions

In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). For an angle



x


{\displaystyle x}
, the sine function is denoted simply as



sin

x


{\displaystyle \sin x}
.More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. More modern definitions express the sine as an infinite series, or as the solution of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year.
The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic, and then from Arabic to Latin. The word "sine" (Latin "sinus") comes from a Latin mistranslation by Robert of Chester of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.

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  1. T

    Why are the paths of our cosmic explorations, pretty?

    TL;DR Summary: Why are the paths of our cosmic explorations, pretty? OK, so I ask a lot of stupid questions. Here's another. Why is this picture, below, pretty? (They are the paths of all our cosmic explorations.) Now, I get the sine, cosine, circles, gravitational attraction, escape...
  2. Mr X

    Derivation or proof of derivative sin (x)

    How do I do this from here without using the derivatives of sin or cos ?
  3. K

    Can a Square Wave Tachometer Drive be Powered by a Sine Waveform?

    I'm a marine engine mechanic, and as engine controls & sensor systems have gotten more complicated with current technology, my shop gets more & more requests for instrumentation & control system repairs. I have a lot of trouble getting technical info from suppliers, so I have been starting to...
  4. A

    Is it possible to solve relative velocity problems without sine law?

    I was able to solve this question successfully by utilizing the sine and cosine law however my instructor said I was only allowed to utilize the vector component method, I am unsure how to complete this question using the vector component method as we have two unknowns(those being the angle of...
  5. A

    Fourier sine and cosine transforms of Heaviside function

    Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. I tried each part of the problem, but I'm only left in terms of limits as x -> infinity of sin or cos...
  6. S

    Find limit involving square of sine

    $$\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n})$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n}-n\pi)$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n}-n\pi)$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi (\sqrt{n^2+n}-n))$$ $$=\lim_{n \rightarrow \infty} \sin^{2}...
  7. Purplepixie

    MHB Closed form solution to sum of sine positive zero-crossings

    Hello, I would like to know, if there's a closed form solution to the following problem: Given a sum of say, 3 sines, with the form y = sin(a.2.PI.t) + sin(b.2.PI.t) + sin(c.2.PI.t) where a,b,c are constants and PI = 3.141592654 and the periods in the expression are multiplication signs, what...
  8. jaumzaum

    I Why does the integral of sine of x^2 from - infinity to + infinity diverge?

    Hello guys. I was trying to evaluate the integral of sine of x^2 from - infinity to + infinity and ran into some inconsistencies. I know this integral converges to sqrt(pi/2). Can someone help me to figure out why I am getting a divergent answer? $$ I = \int_{-\infty}^{+\infty} sin(x^2) dx =...
  9. George Keeling

    B Happy Christmas: Is it Really True?

    Is this really true? It resembles the binomial theorem. I've posted it twice which might be breaking rules. Happy Christmas PF.
  10. B

    I Equation to graph a sine wave that acts like a point on a unit circle

    I need an equation to graph a sine wave that act like a unit circle but only positive numbers. so I need it to be 0 at 0, A at 90 , 0 at 180, A at 270, 0 at 360, and A at 450 and so on and so on... Now I know sin(0) is 0 in degrees and sin(90) 1 and I know if you Square a number is...
  11. S

    I Phase Shift of a Sine Wave

    Hi, I have created a sine wave with the following options: 1.) - changing the period/length in days of the sine wave (Cycle Length in Days) 2.) - calculating the start value of the "dummy" so that the sine wave always starts with -1 (Dummy Start at Cycle Trough) when the phase shift is set...
  12. tworitdash

    A Fourier Transform of an exponential function with sine modulation

    I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time. The time-domain form is, s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}} Here, \mu , \Omega and \lambda are constants. A quick...
  13. MountEvariste

    MHB Definite integral involving sine and hyperbolic sine

    Calculate $\displaystyle \int_0^{\infty} \frac{\sin x}{\cos x + \cosh x}\, \mathrm dx.$
  14. J

    B RC Low Pass Circuit Sine Wave Response

    Hey Everyone, I am trying to gain a level of fundamental understanding of an RC circuit sine wave response through the mathematics and was wondering if someone could help me work it out. Fundamentally a sine wave is represented by the equation y=-ky'' . When a sine wave is used as the input...
  15. anemone

    MHB Linear combination of sine and cosine function

    Hi MHB! I recently came across a problem and I was thinking most likely I was missing something very obvious because I couldn't make sense of what was being asked, and I so wish to know what exactly that I failed to relate. Question: Find the minimum of $6\sin x+8\cos x+5$. Hence, find the...
  16. anemone

    MHB Trigonometric of tangent and sine functions

    Simplify $\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right)$.
  17. penroseandpaper

    Odd and even extension of sine function

    Hi everyone, We've been looking at Fourier series and related topics in online class, touching upon odd and even periodic extensions. However, we haven't looked at what this translates to for sine and cosine functions - only sawtooth and line examples. So, I'm trying to do my own investigation...
  18. anemone

    MHB Can You Meet the Sine Function Challenge?

    Let $a,\,b$ and $c$ be real numbers such that $\sin a+\sin b+\sin c\ge \dfrac{3}{2}$. Prove that $\sin \left(a-\dfrac{\pi}{6}\right)+\sin \left(b-\dfrac{\pi}{6}\right)+\sin \left(c-\dfrac{\pi}{6}\right)\ge 0$.
  19. Theia

    MHB Computing the value of sine function accurately

    Hi all Simple question: How I can compute the value of \(a = \sin \left( 2017 \sqrt[5]{2} \right) \) under following assumptions: No use of advanced numerical libraries is allowed. Only accepted operations are: comparisons, absolute value, addition, subtraction, multiplication and division...
  20. bob012345

    I Exact Expression for Sine of 1 Degree

    For example, $$Sin(15)= \frac {(\sqrt 6 - \sqrt 2)} 4$$ and $$Sin(3)=\frac {(\sqrt 6 (\sqrt 5 -1)(3+\sqrt 3)} {48} -\frac {\sqrt 3 (3-\sqrt 3 )\sqrt{ 5+\sqrt 5 }} {24}$$ What about ##Sin(1)##?
  21. N

    A Sine Dipole formation using two hydrophones

    Hi all. I am trying to do MATLAB simulations for generation of SINE Dipole using two hydrophones spaced distance 'd' apart for signal coming from direction 'DOA'. The MATLAB code is given below. The confusion is that there is constant phase difference of 90 degree b/w SINE Dipole generated using...
  22. F

    I Determining the Equation of a Sine and Cosine Graph that speeds up

    My function needs to speed up towards the left. How do I do this? Green is the graph. Red is my function that needs to match the graph. A = Amplitude = -0.13 H = Phase Shift = 0.1625 V = Vertical Shift = 0.05 P = period = 0.4 B = 2π / P Y = A (Cos(B...
  23. S

    B What is the speed of a photon traveling along the sine function?

    On the image you can see a photon starting at point A at t=0. The photons travels along the sine function and arrives point C. I knot that this takes T=λ/c. But this is the time for a object traveling directly from the origin to point C and not along the sine wave! If the photon travels...
  24. bagasme

    B Derivation of Cosine and Sine Method of Vector Sum

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  25. arcTomato

    Engineering The power spectrum of a sine wave (C language)

    Hi I would like to Derive the power spectrum of sinusoid.I tried like this. But It doesn't work. <Moderator: CODE tags added> #include <stdio.h> #include <math.h> #define pi 3.1415926535 FILE *in_file, *out_file; int main() { dft(); } int dft(int argc, char *argv[]) { char...
  26. Kirkkh

    B Why is sine not used for dot product?

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  27. A

    I How can I go from sine to cosine using exponential numbers?

    ##cos(\omega)## is $$\frac{e^{j \omega } + e^{-j \omega }}{2}$$ ##sin(\omega)## is $$\frac{e^{j \omega } - e^{-j \omega }}{2 j}$$ I also know that ##cos(\omega - \pi / 2) = sin(\omega)##. I've been trying to show this using exponentials, but I can't seem to manipulate one form into the...
  28. I

    A How can you generate a sine wave using integers only?

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  29. karush

    MHB 2.2.206 AP Calculus Practice question derivative of a composite sine

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  30. Robin04

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  31. Benjamin_harsh

    How is the Sine law written for this problem?

    Find the resultant vector of vectors A and B shown in the figure. Solution: By geometry method: Cosine law for the right side triangle. ##R^{2} = 17^{2} + 44^{2} - 2 (17)(14).cos 70^{0}## ##R = 41.39 m/sec## By Sin law, ##\large\frac {R}{Sin 70^0} = \frac {17}{Sin\alpha}## ##sin...
  32. J

    I Are electromagnetic waves sine waves?

    Light is said to consist of photons or electromagnetic waves. I'm not asking which view is correct, what conditions make one view or the other more useful, or advantages and disadvantages of each view. I am assuming the two views are compatible to the extent that the wave character of light can...
  33. Benhur

    Combining Sine Functions: Simplifying with Trigonometry

    Moved from technical forum, so no template is shown Summary: I have the expression sin(2x) + sin(2[x + π/3]) and I have to write this in terms of a single function (a single harmonic, rather saying). But I don't know how to do this, and... it seems a little bit weird for me, because I'm merging...
  34. M

    Uniform convergence of a sine series

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  35. FQVBSina

    A Bessel's Integrals with Cosine or Sine?

    Hello all, This is knowledge needed to solve my take-home final exam but I just want to ask about the definition of Bessel's integrals. This is not a problem on the exam. Wikipedia says the integral is defined as: $$J_n(x) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(xsin(\theta) - n\theta)} \...
  36. Morbidly_Green

    Finding the Sine Representation of an Odd Function Using Fourier Series

    I am attempting to find the sine representation of cos 2x by letting $$f(x) = \cos2x, x>0$$ and $$-\cos2x, x<0$$ Which is an odd function. Hence using $$b_n = \dfrac{2}{l} \int^\pi _0 f(x) \sin(\dfrac{n\pi x}{l})dx$$ I obtain $$b_n = \dfrac{2n}{\pi} \left( \dfrac{(-1)^n - 1}{4-n^2} \right)$$...
  37. E

    Line voltage in 3 phase as single sine wave

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  38. Miles123K

    The sum of this series of the product of 2 sine functions

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  39. E

    MHB Express in terms of sine function of f(x)=sinx+cosx

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  40. Matt Benesi

    B What are cosine and sine functions called in relation to Pi?

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  41. V

    B How to set up an integral to integrate over a sine wave?

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  42. Krushnaraj Pandya

    Is the Sine Rule Valid for Non-Triangular Vector Configurations?

    Homework Statement If magnitudes of vector a,b and c are 1,2,3 respectively and vectors a+b+c=0 then it is obvious that a and b will be in the opposite direction to c therefore cancelling it out but let's assume for a moment that we don't know this and we want to figure this out using the sine...
  43. Saracen Rue

    B Area under a sine integral graph

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  44. opus

    B Cosine or Sine of (angle+angle) always equal to 1?

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  45. opus

    B Adding Sine and Cosine Waves- How to get formula

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  46. T

    MHB Integral of sine = 27/2+ln^2(2)+ln(2)

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  47. D

    What is the mistake in calculating the integral of the absolute sine function?

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  48. T

    Photoelectric effect , Superposition of sine waves

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  49. A

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